The category of posets
Here is a fact that should be much more widely known than it is. The category of posets is isomorphic (not just equivalent) to the category of $T_0$ Alexandrof spaces. A topological space is said to be Alexandrof if arbitrary (not just finite) intersections of open sets are open. For example, every finite topological space is an Alexandrov space. Finite spaces are fascinating and the connection with algebraic topology is very close: finite spaces have associated finite simplicial complexes. Don't just look categorically: that takes the fun out of it! There are notes on my web page and there is a book by Barmak.
Here are some basic remarks and examples: (Caution. This answer refers to preorders; but many of the remarks also apply to partially ordered sets aka posets)
Many concepts of category theory have a nice illustration when applied to preorders; but also the other way round: Many concepts familiar from preorders carry over to categories (for example suprema motivate colimits; see also below).
This is partially justified by the following observation: An arbitrary category is a sort of a preorder but where you have to specify in addition a reason why $x \leq y$, in form of an arrow $x \to y$. The axioms for a category tell you: For every $x$ there is a distinguished reason for $x \leq x$, and whenever you have a reason for $x \leq y$ and for $y \leq z$, you also get a reason for $x \leq z$.
A preorder is a category such that every diagram commutes.
In a preorder, the limit of a diagram is the same as the infimum of the involved objects. Similarly, a colimit is just a supremum. The transition morphisms don't matter.
When $f^* : P \to Q$ is a cocontinuous functor between preorders, where $P$ is complete, then $f^*$ has a right adjoint $f_*$; you can write it down explicitly: $f_*(q)$ is the infimum of the $p$ with $f^*(p) \leq q$. This construction motivates the General Adjoint Functor Theorem. In this setting we only have to add the solution set condition, so that the a priori big limit can be replaced by a small one and therefore exists.
Let $f : X \to Y$ be a map of sets. Then the preimage functor $\mathcal{P}(Y) \to \mathcal{P}(X)$ between the power sets is right adjoint to image functor $\mathcal{P}(X) \to \mathcal{P}(Y)$. Every cocontinuous monoidal functor $\mathcal{P}(Y) \to \mathcal{P}(X)$ arises this way.
The inclusion functor $\mathrm{Pre} \to \mathrm{Cat}$ has a left adjoint: It sends every category to its set of objects with the order $x \leq y$ if there is a morphism $x \to y$. In particular, it preserves all limits. In fact, it creates all limits, and limits in $\mathrm{Cat}$ are constructed "pointwise". Thus, the same is true for limits in $\mathrm{Pre}$ (which one could equally well see directly). For example, the pullback of $f : P \to Q$ and $g : P' \to Q$ is the pullback of sets $P \times_Q P'$ equipped with the order $(a,b) \leq (c,d)$ iff $a \leq c$ and $b \leq d$. If we apply this to difference kernels, we see that $f : P \to Q$ is a monomorphism iff the underlying map of $f$ is injective.
The forgetful functor $\mathrm{Pre} \to \mathrm{Set}$ creates coproducts: Take the disjoint union $\coprod_i P_i$ and take the order $a \leq b$ iff $a,b$ lie in the same $P_i$, and with respect to that preorder we have $a \leq_i b$.
The construction of coequalizers seems to be more delicate; see this SE discussion.
I don't have a reference for all these observations, but they are easy. A general reference for basic category-theoretic constructions (and it surely says something about preorders and posets) is the book "Abstract and Concrete Categories - The Joy of Cats" by Adamek, Herrlich, Strecker which you can find online.
EDIT: Here is something not so basic: Sefi Ladkani studied the notion of derived equivalent posets. Two posets $X,Y$ are called (universally) derived equivalent if for some specific (every) abelian category $\mathcal{A}$ the diagram categories $\mathcal{A}^X$, $\mathcal{A}^Y$ are derived equivalent.
Probably not what you are looking for, but:
There is this recent paper by George Raptis:
Homotopy theory of posets, 2010
discussing about model category structures on the category of posets.